Negative moments of the gaps between consecutive primes

Abstract

We derive heuristically approximate formulas for the negative $k$--moments $M_{-k}(x)$ of the gaps between consecutive primes$<x$ represented directly by $\pi (x)$ --- the number of primes up to $x$. In particular we propose an analytical formula for the sum of reciprocals of gaps between consecutive primes $<x:M_{-1}(x)\sim \frac{{\pi }^2(x)}{x-2\pi (x)}\log \left(\frac{x}{2\pi (x)}\right)\sim x\log \log (x)/{\log }^2(x)$. We illustrate obtained results by the enormous computer data up to $x=4\times 10^{18}$.

Many improvements and changes. Apparently for the first time the analytical formula for the sum of reciprocals of gaps between consecutive primes is presented

References in corpus (1)

• Nearest neighbor spacing distribution of prime numbers and quantum chaos